Rational Points on Elliptic Curves: x^3-7y^3=1, v^2=u^3-21168

[2003.12.23]x^3-7y^3=1, v^2=u^3-21168の有理点

x^3-7y^3=1の有理点を計算するプログラム(pari/gp)

■Diophantus方程式
     C: x³-7y³ = 1 ----- (1)
で表される楕円曲線の有理点(x,y)を求める。

[case i] y=0の場合
(1)でy=0とすると、x³=1より、x=1を得る。

[case ii] y!=0の場合
(1)の両辺をy³で割ると、
     (x/y)³-7 = (1/y)³
     (x/y)³+(-1/y)³ = 7 ----- (2)
を得る。
よって、(x/y,-1/y)は、楕円曲線
     C₂: x³+y³ = 7 ----- (3)
の有理点である。

■曲線Cは、整点(1,0), (2,1)を持つ。
また、曲線Cの整点は、(1,0), (2,1)に限ることが分かる。
pari/gpでThue方程式(1)を解くと、以下のようになる。

gp>  th=thueinit(x^3-7)
time = 519 ms.
%13 = [x^3 - 7, [Mat(3), Mat([0, 2, 1, 1, 1, 2, 1, 2, 2]), [-2.441056470389030485120336455 + 9.424777960769379715387930149*I; 2.441056470389030485120336455 + 7.304690385164919035485815990*I], [-16.71138261976191981732515107 + 9.424777960769379715387930149*I, -2.441056470389030485120336455 + 0.E-57*I, -0.9150925090658454936608853936 + 0.E-57*I, -1.446915436617573493185553569 + 3.141592653589793238462643383*I, -0.6883479324759123022010441127 + 0.E-57*I, -4.882112940778060970240672911 + 6.283185307179586476925286766*I, 2.893493921667412223871043315 + 6.283185307179586476925286766*I, 0.4164172377127375918074907666 + 0.E-57*I, -1.150390173465463910676126977 + 0.E-57*I, -5.873191343057422126401181013 + 6.283185307179586476925286766*I; 16.71138261976191981732515107 + 10.34467336044623014633928300*I, 2.441056470389030485120336455 + 9.399085487558114527794244912*I, 0.9150925090658454936608853936 + 0.7930770208573159086918341234*I, 1.446915436617573493185553569 + 11.40497600267959792086356030*I, 0.6883479324759123022010441127 + 7.688399604690998075250655986*I, 4.882112940778060970240672911 + 2.043010155970665117121058447*I, -2.893493921667412223871043315 + 1.412550356250449529211152063*I, -0.4164172377127375918074907666 + 11.20393525618899268634710750*I, 1.150390173465463910676126977 + 11.69747797333210648070325166*I, 5.873191343057422126401181013 + 7.887628069845086533888385121*I], [[2, [3, 1, 0]~, 1, 1, [1, 1, 1]~], [2, [1, 1, 1]~, 1, 2, [1, 1, 0]~], [3, [-1, 1, 0]~, 3, 1, [1, 1, 1]~], [5, [2, 1, 0]~, 1, 1, [-1, -2, 1]~], [7, [0, 1, 0]~, 3, 1, [0, 0, 1]~], [11, [5, 1, 0]~, 1, 1, [3, -5, 1]~], [17, [3, 1, 0]~, 1, 1, [-8, -3, 1]~], [19, [-6, 1, 0]~, 1, 1, [-2, 6, 1]~], [19, [-4, 1, 0]~, 1, 1, [-3, 4, 1]~], [19, [10, 1, 0]~, 1, 1, [5, 9, 1]~]]~, [1, 5, 4, 6, 3, 2, 9, 7, 10, 8], [x^3 - 7, [1, 1], -1323, 1, [[1, 1.912931182772389101199116839, 3.659305710022971517238073310; 1, -0.9564655913861945505995584197 - 1.656646999972302077004874245*I, -1.829652855011485758619036655 + 3.169051705093345866738808850*I], [1, 2; 1.912931182772389101199116839, -1.912931182772389101199116839 + 3.313293999944604154009748490*I; 3.659305710022971517238073310, -3.659305710022971517238073310 - 6.338103410186691733477617701*I], [3, 0.E-96, 0.E-96; 0.E-96, 10.97791713006891455171421993, -7.49068216 E-96; 0.E-96, -7.49068216 E-96, 40.17155483822017112518145363], [3, 0, 0; 0, 0, 21; 0, 21, 0], [21, 0, 0; 0, 21, 0; 0, 0, 3], [-441, 0, 0; 0, 0, -63; 0, -63, 0], [441, [0, 63, 0]~]], [1.912931182772389101199116839, -0.9564655913861945505995584197 - 1.656646999972302077004874245*I], [1, x, x^2], [1, 0, 0; 0, 1, 0; 0, 0, 1], [1, 0, 0, 0, 0, 7, 0, 7, 0; 0, 1, 0, 1, 0, 0, 0, 0, 7; 0, 0, 1, 0, 1, 0, 1, 0, 0]], [[3, [3], [[2, 1, 1; 0, 1, 0; 0, 0, 1]]], 2.441056470389030485120336455, 0.9224868950882813690, [2, -1], [x - 2], 187], [Mat(1), [[0, 0]], [[-16.71138261976191981732515107 + 9.424777960769379715387930149*I, 16.71138261976191981732515107 + 10.34467336044623014633928300*I]]], 0], [1.912931182772389101199116839 + 0.E-48*I, -0.9564655913861945505995584197 + 1.656646999972302077004874245*I, -0.9564655913861945505995584197 - 1.656646999972302077004874245*I]~, [1.627370980259353656746890970]~, [-0.08706881722761089880088316045 + 0.E-48*I; -2.956465591386194550599558419 + 1.656646999972302077004874245*I; -2.956465591386194550599558419 - 1.656646999972302077004874245*I], Mat(-0.4096586916895987582123542806), [0.3643678443740347084716868094, 3.313293999613274754015288075, 0.6486367163517711017017842478, 0.6036288962082563555692969239, 7.902542435962593014830339723 E-49, 7]]
gp>  thue(th,1)
time = 18 ms.
%14 = [[1, 0], [2, 1]]

■双有理変換φ:(x,y)→(84y/(x-1), 252(x+1)/(x-1))[逆変換は、φ^-1:(u,v)→(6u/(v-252),(v+252)/(v-252))]によって、曲線Cは、楕円曲線
E: v² = u³ - 21168 ----- (4)
に写される。
ただし、φ(1,0)=Oとする。ここで、O=[0:1:0]は、曲線Eの無限遠点である。

■楕円曲線Eのねじれ点群E_tors(Q)は、自明な群である。

pari/gpで計算すると、以下のようになる。
E_tors(Q) = { O }

[pari/gpでの計算結果]

gp>  e=ellinit([0,0,0,0,-21168])
time = 221 ms.
%3 = [0, 0, 0, 0, -21168, 0, 0, -84672, 0, 0, 18289152, -193572384768, 0, [27.66261775235067941804817658, -13.83130887617533970902408829 + 23.95652970871407777593509002*I, -13.83130887617533970902408829 - 23.95652970871407777593509002*I]~, 0.4617622333190995379444907679, 0.2308811166595497689722453839 + 0.3998978245625773463769204533*I, -3.927994178295644033866426142 + 0.E-28*I, -1.963997089147822016933213071 - 10.20522823296422824587520533*I, 0.1846577125694651747411467103]
gp>  elltors(e,1)
time = 256 ms.
%4 = [1, [], []]

■楕円曲線EのMordell-Weil群E(Q)をCremonaのmwrank3で計算すると、rankは1であり、その生成元は
P(84,756)
である。

E(Q) = Z

[mwrank3での計算結果]

bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,0,0,0,-21168]    

Curve [0,0,0,0,-21168] :        Working with minimal curve [0,0,1,0,-331]
        [u,r,s,t] = [2,0,0,4]
No points of order 2
Basic pair: I=0, J=571536
disc=-326653399296
2-adic index bound = 2
By Lemma 5.1(a), 2-adic index = 1
2-adic index = 1
One (I,J) pair
Looking for quartics with I = 0, J = 571536
Looking for Type 3 quartics:
Trying positive a from 1 up to 15 (square a first...)
(1,0,-126,756,-1323)    --nontrivial...(x:y:z) = (1 : 1 : 0)
Point = [21 : 94 : 1]
        height = 0.298091926737864
Rank of B=im(eps) increases to 1
(1,0,-42,28,-147)       --trivial
(9,-12,-36,36,-24)      --trivial
Trying positive a from 1 up to 15 (...then non-square a)
(2,2,-378,2646,-5292)   --trivial
Trying negative a from -1 down to -23
(-3,0,0,84,0)   --trivial
(-3,-3,117,471,498)     --trivial
(-3,-4,-42,0,49)        --trivial
(-6,6,-18,66,-12)       --trivial
(-10,-13,-36,16,16)     --trivial
Finished looking for Type 3 quartics.
Mordell rank contribution from B=im(eps) = 1
Selmer  rank contribution from B=im(eps) = 1
Sha     rank contribution from B=im(eps) = 0
Mordell rank contribution from A=ker(eps) = 0
Selmer  rank contribution from A=ker(eps) = 0
Sha     rank contribution from A=ker(eps) = 0
Rank = 1
Points generating E(Q)/2E(Q):
Point [84 : 756 : 1], height = 0.298091926737864
After descent, rank of points found is 1
Transferring points back to original curve [0,0,0,0,-21168]

Generator 1 is [84 : 756 : 1]; height 0.298091926737864

The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
 (modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 0.298091926737864

 (4.1 seconds)
Enter curve: [0,0,0,0,0]

bash-2.05a$

■pari/gpで、楕円曲線E: v² = u³-21168の有理点をいくつか計算すると、以下のようになる。

gp> read("de5.gp")
time = 36 ms.
gp> v=ratpointE(20,50);
[0]
[84, -756]
[84, 756]
[28, -28]
[28, 28]
[57, 405]
[57, -405]
[1708, 70588]
[1708, -70588]
[116004/841, -39350556/24389]
[116004/841, 39350556/24389]
[27721/900, -2422981/27000]
[27721/900, 2422981/27000]
[114614724/2725801, 1037753178084/4500297451]
[114614724/2725801, -1037753178084/4500297451]
[2713877068/6355441, 141359895391652/16022066761]
[2713877068/6355441, -141359895391652/16022066761]
[149447123337/555969241, -57742362828488115/13109198733539]
[149447123337/555969241, 57742362828488115/13109198733539]
[84322757779708/2278526851441, -590895790409891584828/3439388433186309239]
[84322757779708/2278526851441, 590895790409891584828/3439388433186309239]
[220549795975000884/6575626861077121, 68625708386742575986356684/533219153739563539321919]
[220549795975000884/6575626861077121, -68625708386742575986356684/533219153739563539321919]
[4012727988243653281/21135012934899600, 8025773933749949127319297679/3072584086560958062456000]
[4012727988243653281/21135012934899600, -8025773933749949127319297679/3072584086560958062456000]
[562111677460675323908724/761281054067112882241, -421427132117100391705419895922761164/21004750755406324415561430299039]
[562111677460675323908724/761281054067112882241, 421427132117100391705419895922761164/21004750755406324415561430299039]
[247429364759793299249325628/5136161837587549027575121, -3504259440049846536097994959514795486852/11640135865015873519213570682197354919]
[247429364759793299249325628/5136161837587549027575121, 3504259440049846536097994959514795486852/11640135865015873519213570682197354919]
[412534943441553714652431091497/14182520436233809792840784521, 99099754094473706533925126377870303982393475/1689001856938973862814282872049688792489819]
[412534943441553714652431091497/14182520436233809792840784521, -99099754094473706533925126377870303982393475/1689001856938973862814282872049688792489819]
[17335171645849514748648722616342028/161987707391821603304864586855121, 2262605788107145965796142416434074575787869659070052/2061688689353859226219516908871239005964319234919]
[17335171645849514748648722616342028/161987707391821603304864586855121, -2262605788107145965796142416434074575787869659070052/2061688689353859226219516908871239005964319234919]
[1967043169190114034251714327163547560324/312514315806877991963743913861590441, -87241036813119140904513592028670682596823505601717564952164/174704815048648993383479552130320621145591111486188011]
[1967043169190114034251714327163547560324/312514315806877991963743913861590441, 87241036813119140904513592028670682596823505601717564952164/174704815048648993383479552130320621145591111486188011]
[690228439560379148362187673583820342783881/10171203196117412546765818670633926184100, -553679925496312202567176993727166502909803010808757568040610779/1025790082289359415824413119483130405895727033382236247111000]
[690228439560379148362187673583820342783881/10171203196117412546765818670633926184100, 553679925496312202567176993727166502909803010808757568040610779/1025790082289359415824413119483130405895727033382236247111000]
time = 120 ms.

■pari/gpで、楕円曲線C: x³-7y³ = 1の有理点をいくつか計算すると、以下のようになる。

gp>  ratpointC(v)
[1, 0]
[2, 1]
[1/2, -1/2]
[-5/4, -3/4]
[-4/5, -3/5]
[17/73, -38/73]
[73/17, 38/17]
[1256/1265, -183/1265]
[1265/1256, 183/1256]
[90271/65882, 40049/65882]
[65882/90271, -40049/90271]
[-9226981/4381019, -4989780/4381019]
[-4381019/9226981, -4989780/9226981]
[-191114642/4309182809, -2252725111/4309182809]
[-4309182809/191114642, -2252725111/191114642]
[2452184545855/2596383146704, -733037580903/2596383146704]
[2596383146704/2452184545855, 733037580903/2452184545855]
[6782875656593327/6048760527515143, 2349209147442082/6048760527515143]
[6048760527515143/6782875656593327, -2349209147442082/6782875656593327]
[-26028958492372169876/4925537406304613275, -13637510581130984157/4925537406304613275]
[-4925537406304613275/26028958492372169876, -13637510581130984157/26028958492372169876]
[-130447457054816341116601/402771696684747198205318, -212909948416834044152039/402771696684747198205318]
[-402771696684747198205318/130447457054816341116601, -212909948416834044152039/130447457054816341116601]
[7251482743936587695580385679/8800065123563310559058209679, -3500194922525734297362237960/8800065123563310559058209679]
[8800065123563310559058209679/7251482743936587695580385679, 3500194922525734297362237960/7251482743936587695580385679]
[846667320054489653686788445154998/825662569299083329271227014855959, 184635765918862591890633965260919/825662569299083329271227014855959]
[825662569299083329271227014855959/846667320054489653686788445154998, -184635765918862591890633965260919/846667320054489653686788445154998]
[114956672822032976123925263775509445115/10195450036890114451003127635733250844, 60080538052780477684491099282986517117/10195450036890114451003127635733250844]
[10195450036890114451003127635733250844/114956672822032976123925263775509445115, -60080538052780477684491099282986517117/114956672822032976123925263775509445115]
[-36280968206016411877252684153183474636115657/58303135782566124440347156681599097743314207, -32752630895557411353284157726995949516531922/58303135782566124440347156681599097743314207]
[-58303135782566124440347156681599097743314207/36280968206016411877252684153183474636115657, -32752630895557411353284157726995949516531922/36280968206016411877252684153183474636115657]
[31126075685535240014086145632116470469372521640544/49681273889719973050061797811957621523051394754815, -23639179535179492707527243864413154701817508224217/49681273889719973050061797811957621523051394754815]
[49681273889719973050061797811957621523051394754815/31126075685535240014086145632116470469372521640544, 23639179535179492707527243864413154701817508224217/31126075685535240014086145632116470469372521640544]
[173184647671649604069139343007554609907444830479785881609/173009942856600955075755863455424289286299239368299693598, 13090901748307774812292602467927659434861003633157453831/173009942856600955075755863455424289286299239368299693598]
[173009942856600955075755863455424289286299239368299693598/173184647671649604069139343007554609907444830479785881609, -13090901748307774812292602467927659434861003633157453831/173184647671649604069139343007554609907444830479785881609]
[812179026233230775354929099836915365195526223221081102312582779/295180824759393629779424887617417640624079798396434033768638779, 417667098471910761530408052381574303222775864734610799595677060/295180824759393629779424887617417640624079798396434033768638779]
[295180824759393629779424887617417640624079798396434033768638779/812179026233230775354929099836915365195526223221081102312582779, -417667098471910761530408052381574303222775864734610799595677060/812179026233230775354929099836915365195526223221081102312582779]
time = 9 ms.

[参考文献]

[1]Joseph H.Silverman, John Tate(著), 足立恒雄, 木田雅成, 小松啓一, 田谷久雄(訳), "楕円曲線論入門", シュプリンガー・フェアラーク東京, 1995, ISBN4-431-70683-6, {3900円}.
[2]Joseph H. Silverman, "The Arithmetic of Elliptic Curves", GTM 106, Springer-Verlag New York Inc., 1986, ISBN0-387-96203-4.

Last Update: 2005.06.12

H.Nakao